Toward incompatible quantum limits on multiparameter estimation

Achieving the ultimate precisions for multiple parameters simultaneously is an outstanding challenge in quantum physics, because the optimal measurements for incompatible parameters cannot be performed jointly due to the Heisenberg uncertainty principle. In this work, a criterion proposed for multiparameter estimation provides a possible way to beat this curse. According to this criterion, it is possible to mitigate the influence of incompatibility meanwhile improve the ultimate precisions by increasing the variances of the parameter generators simultaneously. For demonstration, a scheme involving high-order Hermite-Gaussian states as probes is proposed for estimating the spatial displacement and angular tilt of light at the same time, and precisions up to 1.45 nm and 4.08 nrad are achieved in experiment simultaneously. Consequently, our findings provide a deeper insight into the role of Heisenberg uncertainty principle in multiparameter estimation, and contribute in several ways to the applications of quantum metrology.


Supplementary Note 1: Derivation of quantum multiparameter estimation criterion
In this section, we would like to give the derivation details of quantum multiparameter criterion (QMEC).

Trade-off precision relation of incompatible parameters
Recently, Xiaoming Lu and Xiaoguang Wang proposed a trade-off relation for the measurement inaccuracies of different parameters[1] by incorporating the Heisenberg's uncertainty principle (HUP) and Ozawa's uncertainty relation [2,3]. Here we would like to begin with the measurement uncertainty relations. LetÂ andB be two Hermitian operators standing for the ideal observables we intend to measure. Due to the HUP, when [Â,B] ̸ = 0, these two observables can not be jointly measured. To approximately the joint measurement of the non-commutative observ-ablesÂ andB, we can dilate the system by adding an ancilla η, where another pair of commuting observablesÂ and B acting on the dilated system are measured [4]. Then the measurement errors for the ideal observablesÂ andB in the quantum state ρ is defined by Ozawa as: Combining with the uncertainties of observablesÂ andB in the quantum state ρ: Ozawa derives the inequality: where [Â,B] ≡ÂB −BÂ, it is a universally joint-measurement error relation. Ozawa's inequality was recently strengthened by Branciard [3]: which is tight when ρ is a pure state. Lu and Wang takeL i →Â andL j →B, whereL i andL j are respectively the symmetric logarithmic derivatives (SLD) for the parameters g i and g j in the parameterized quantum state ρ g [5], then the uncertainties can be calculated as: where Q ii and Q jj are respectively the i-th and j-th diagonal elements of quantum Fisher information matrix (QFIM) Q. They also takeL i →Â andL j →B, which are constructed from the SLD of the output state λ tr Π λ ρ g |λ⟩⟨λ| under a set of POVMΠ = Π λ Π λ ≥ 0, λΠ λ =Î , then the measurement errors ϵ A and ϵ B can be replaced by[1]: where F jj is the classical Fisher information matrix under the given POVMΠ, and R = Q−F is named as information regret. Substituting Eq. (5) and Eq. (6) into the Branciard's inequality, Lu and Wang derive the trade-off relation between the information regrets of different parameters[1]: where C ij ≡ 1 2 tr [L i ,L j ]ρ g . This trade-off relation is tight for pure state ρ g = |ψ g ⟩⟨ψ g |. For pure state, it is easy to derive that C ij = 2C ij , where C is the Berry curvature (BC). Furthermore, combining with the classical Cramér-Rao where T is the quantum geometric tensor (QGT), which can be calculated as: whereĤ i = i ∂ ∂giÛ † (g) Û (g) is the generator of parameter g i , and ⟨·⟩ denotes for ⟨ψ| · |ψ⟩. Then the QFIM and BC can be respectively given by: where {Ĥ i ,Ĥ j } =Ĥ iĤj +Ĥ jĤi , and Therefore, the QMEC can be calculated by: where

QMEC of post-selected weak measurement scheme
In the multiparameter weak measurement scheme, every unknown parameter g j ∈ g = (g 1 , g 2 , . . . , g n ) is coupled to the pointer via a corresponding translation operatorΩ j during the weak interaction procedure [6]. Then the weak interaction process is described by an impulse Hamiltonian: whereÂ is an hermitian operator on the two-level system. Then the evolution operator in multiparameter weak measurement scheme isÛ The initial state of the whole system is denoted as |Ψ i ⟩ = |i⟩|ψ i ⟩, where |i⟩ is the pre-selected state of the two-level system, and |ψ i ⟩ is the initial state of pointer. After the weak interaction process, the two-level system is post-selected by state |f ⟩, then the final state of the whole system is where A w = ⟨f |Â|i⟩/⟨f |i⟩ is the weak value, and | ψ f ⟩ is the final state of pointer. Here, we we calculate the approximate results based on the weak interaction condition: g j ≪ 1, j = 1, 2, . . . , n, and only the first-order small quantity is retained. Then the final pointer state |ψ f ⟩ can be normalized by a normalization factor: where ⟨·⟩ i denotes for ⟨ψ i | · |ψ i ⟩, the normalized final pointer state is: Thus, it is easy to determine that the QGT in multiparameter weak measurement scheme is: The corresponding QFIM and BC is then calculated as: and Then the QMEC of post-selected weak measurement scheme can be calculated as: Moreover, we can approximately take |ψ f ⟩ ≈ 1 − iA w n j=1 g jΩj |ψ i ⟩ in this case. In this work, we concentrate on the incompatible parameters generated by operatorsP andX, then the corresponding QMEC can be calculated as: Supplementary Note 2: Simultaneous quantum limits and energy level of probe state In this section, we would like to clarify that the attainability of simultaneous quantum limits for incompatible parameters is independent of the energy level of initial probe state, where two situations are involved: (1) two Gaussian states with different energy levels; and (2) Gaussian state and n-order HG state with the same energy level. Here, we still concern with the simultaneous measurement of incompatible parameters g 1 and g 2 generated by the momentum operatorP and the position operatorX where the unitary parameterization isÛ (g) = exp −ig 1P − ig 2X . First, considering the time-independent solution ψ n (x, ω) for the Schrödinger equation of harmonic oscillators: For energy level E n = n + 1 2 ℏω, the corresponding eigenket is |n, ω⟩ = dx ψ n (x, ω)|x⟩, which corresponds to the n-order HG state. And |0, ω⟩ corresponds to the Gaussian state with energy level E n = 1 2 ℏω. It is easy to obtain that:
It is easy to determine that Q(g 1 , |0, ω 1 ⟩) < Q(g 1 , |0, ω 2 ⟩), but Q(g 2 , |0, ω 1 ⟩) > Q(g 2 , |0, ω 2 ⟩), which means that increasing the energy level of Gaussian state will improve the quantum precision limit of parameter g 1 , but reduce the quantum precision limit of parameter g 2 . However, the attainability of simultaneous quantum limits for parameters g 1 and g 2 depends on the QMEC S 12 , where the Gaussian states |0, ω 1 ⟩ and |0, ω 2 ⟩ give the same results: Thus, the curve properties of the trade-off bounds with the Gaussian states |0, ω 1 ⟩ and |0, ω 2 ⟩ are the same. This result leads to the same attainability of simultaneous quantum limits for parameters g 1 and g 2 with the Gaussian states |0, ω 1 ⟩ and |0, ω 2 ⟩. For illustration, we have plotted the trade-off bounds with the Gaussian states |0, ω 1 ⟩ and |0, ω 2 ⟩ in Supplementary Figure 1.
Although the Gaussian state and the HG n state are on the same energy level, their corresponding QMECs are difference, which are calculated as: S 12 (|0, ω 1 ⟩) = 1, S 12 (|n, ω 2 ⟩) = 1 (2n + 1) 2 (33) Thus, the attainability of simultaneous quantum limits for the incompatible parameters g 1 and g 2 with the HG n state |n, ω 2 ⟩ is improved (2n + 1) 2 -fold than that with the Gaussian state |0, ω 1 ⟩, though they are on the same energy level. For illustration, we have plotted the trade-off bounds with the Gaussian states |0, ω 1 ⟩ and the HG n state |n, ω 2 ⟩ separately in Supplementary Figure 2.
From these two examples, we can conclude that the QMEC S 12 for evaluating the attainability of simultaneous quantum limits is independent of the energy level of initial probe state. The attainability of simultaneous quantum limits depends on the uncertainty properties of the initial probe state regarding the parameters' generators.

Generation method of HG beams
Traditionally, mode cleaner cavity is necessary for generating high-order HG beams [7,8]. However, mode cleaner cavity is usually difficult to setup and control in experiment. In this work, we generate the high-order HG beams by a SLM and 4-f spatial filter system [9], which is easier to implement in experiment. And the generation scheme is illustrated in Supplementary Figure 3. In this scheme, the light beam from a 780 nm DBR laser was expanded to a 8.6 mm-width Gaussian beam by a fiber coupler. The complex amplitude of expanded Gaussian beam is denoted as: A in (x, y) exp [iϕ in (x, y)]. Then inputting this light into a SLM, where the phase map H(x, y) is displayed. The output amplitude of SLM can be denoted as: Here, we denote the relative phase as: ϕ r = ϕ out − ϕ in + ϕ g , where ϕ g is the grating phase, and the relative amplitude is A r = A out /A in . To filter the target light, we let Based on the Bessel expansion formula: where J q [·] is the q-order Bessel function. Thus, we have the amplitude of the 1st-order diffraction beam is A in · J 1 [f (A r )] exp(iϕ out ). The mapping function f (·) can be easily derived as: where J −1 1 (A r ) is the inverse function of 1st-order Bessel function. By employing a 4-f spatial filter system with an aperture at the 1st-order diffraction point, we can generate any target beam amplitude T (x, y) = A out exp(iϕ out ) with the displayed phase map H(x, y) = J −1 1 (A r ) sin(ϕ r ) on SLM.

Phase lock in the interferometer
In our experiment, the incompatible parameters were simultaneously generated in a polarized Mach-Zehnder interferometer (MZI). To stabilize the relative phase of two optical path in the MZI, a lock-in amplifier was employed, as is illustrated in Supplementary Figure 4. Figure 4. Schematic of lock-in amplifier in the MZI. The second output port of the MZI is used to monitor the relative phase of the interferometer. The error signal is feedback to the PZT driven mirror to control the relative phase in the interferometer.
Here, we would like to briefly introduce how to lock the relative phase in our polarized MZI. The input polarization state is denoted as |in⟩ = 1 √ 2 (|H⟩ + |V ⟩). We exerted a cosine modulation β cos(2πf t) on the |V ⟩ path in the MZI via a PZT driving mirror, where β is the modulation depth. Denoting the relative phase in the two optical path as ϕ, then the two output polarization states are separately calculated as: The output state |out⟩ 2 was projected to the polarization state |+⟩ = 1 √ 2 (|H⟩ + |V ⟩) via a polarizer, and then detected by a photodetector (PD) with probability: Re e iϕ+iβ cos(2πf t) The detected optical power in PD I det ∝ P det , then the detected power at frequency f is I In the lock-in amplifier, the detected optical signal was demodulated by multiplying a cosine signal at frequency f , and then a low-pass filter was employed to get the signal I (f ) det . Finally, a PID controller and a high-voltage amplifier (HVA) were used to feed bake the error signal to the PZT driving mirror for stabilizing the relative phase ϕ = 0.
In Supplementary Figure 5, we illustrated the detected ellipticities of polarization state |out⟩ 1 . The orange line stands for the ellipticity of output polarization with lock-in amplifier, and the blue line stands for the ellipticity of output polarization without lock-in amplifier. Obviously, the relative phase was well stabilized in our interferometer with lock-in amplifier.

Projective measurement
In our experimental scheme, the rotation signal was finally detected by projective measurement [10], which is illustrated in Supplementary Figure 6. Suppose that the input light field on SLM is g(x, y), and the modulation light field on SLM is h(x, y) (modulation method is same as the generation method of HG beams). The input field and the modulation field are simply combined as g(x, y)h(x, y) on SLM, and a Fourier lens transfer this filed to which is spatially filtered by a SMF coupled to an APD, the coupling efficiency into the fiber is given as: where w f is the field width of fiber mode. In our experiment, w f = 4.6 µm, which is much smaller of size scale than the features of f (u, v). (The focal length of the Fourier lens is 10 cm, which transfer the waist width of 500 µm HG00 beam to nearly 50 µm.) Therefore, we have f (u, v) exp (u 2 + v 2 )/w 2 f dudv ≈ f (0, 0) exp (u 2 + v 2 )/w 2 f dudv, which leads to In this experiment, we implemented the projection measurementΠ 1 = |u ⊥ X ⟩⟨u ⊥ X | by displaying the phase map of spatial function on the SLM. Similarly, the projection measurementΠ 2 = |u ⊥ P ⟩⟨u ⊥ P | was implemented by displaying the phase map of spatial function on the SLM.

Supplementary Note 4: Supplement of experimental results
In practice, the amplitudes of displacement and tilt signals are reflected by the levels of the peak power at 2 kHz on the spectrum analyzer. When displayingΠ i (i = 1, 2) projection measurement on the SLM, the practical detected signal-to-noise ratio in the experiment is obtained from: where V sni ∝ δI i is the level of detected shot-noise floor and V si ∝ I (2 kHz) i is the level of signal at 2 kHz. The shot-noise level is calculated by V sni = V ni − V en , where V ni is the level of total noise floor on the spectrum analyzer, and V en is the level of electrical noise floor on the spectrum analyzer. The signal level is calculated by V si = V pi − V ni , where V pi is the level of the peak power at 2 kHz on the spectrum analyzer. Generally, the electrical levels V pi and V ni vary with the modes of experimental HG beams, and the electrical noise floor V en = 60.61 µV is a constant value in our experiment. And the constant electrical noise floor V en = 60.61 µV was detected without inputting light on the APD.
In Supplementary Figure 7, we illustrated the spectrums of detected optical power in APD with two different projection measurements and HG modes from 1 to 5. Moreover, we exerted an 1 V peak-to-peak level and 2 kHz Supplementary Figure 7. Spectrums of detected optical power on APD of two different projection measurements with HG modes from 1 to 5. The first line in each figure is the noise floor without inputting light on the APD. The peak-to-peak level of driving signal on the PZT chips is 1 V, and the frequency of driving signal is 2 kHz. a Spectrums with projective measurementΠ 1 = |u ⊥ X ⟩⟨u ⊥ X |. b Spectrums with projective measurementΠ 2 = |u ⊥ P ⟩⟨u ⊥ P |. b Measured from the level at 2 kHz in the spectrum analyzer, which corresponds to 1 V peak-to-peak level and 2 kHz driving signal on PZT chips. c Corresponding to the electrical level of noise floor without light inputting in the APD, which is a constant value in the experiment.

Supplementary
driving signal on the PZT chips of the mirror in MZI. The measured noise floors and signal levels in the spectrum analyzer are illustrated in Supplementary Table I.
To determine the minimum detectable signal, we increased the driving voltages of PZT chips from 0 V to 0.5 V, the detected signal levels at 2 kHz of two projection measurements with HG modes from 1 to 5 are illustrated in FIg.  8. From the detected signal levels in spectrum analyzer, we can fit a linear relation between the driving voltage and the signal level at 2 kHz: V si = r i V D , where V D is the driving voltage of PZT chips, and r i is the linear coefficient calculated from Supplementary Figure 8. When SNR i = 1, V si = V sni . Therefore, the minimum detected value of parameters g 1 and g 2 in our experiment can be obtained by δg min 1det = g u 1 (V n1 − V en )/r 1 and δg min 2det = g u 2 (V n2 − V en )/r 2 , where g u 1 and g u 2 are the unit signal amplitudes (which is given in the Method part of main text).  Figure 8. Detected signal levels at 2 kHz of two projection measurements with HG modes from 1 to 5. The peak-to-peak level of driving signals on PZT chips increase from 0 V to 0.5 V. a Detected signal levels with projective measurementΠ 1 = |u ⊥ X ⟩⟨u ⊥ X |. b Detected signal levels with projective measurementΠ 2 = |u ⊥ P ⟩⟨u ⊥ P |.